Crystal structure note
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1) The document discusses calculating the density of silicon using its diamond crystal structure and lattice constant of 5.43 Angstrom. The density is calculated to be 2342.38 kg/m3. 2) The number density of silicon is calculated to be approximately 5×1022 atoms/cm3 based on its lattice constant and unit cell volume. 3) An equation is presented for the Liedemann model of melting, which states that melting occurs when the amplitude of atomic vibrations exceeds a critical fraction of the interatomic spacing. The melting temperature depends on factors like the force constant, lattice constant, and elastic modulus.
Crystal structure note
- 1. - 1 - Crystal Structure Example) Calculate silicon density. Lattice constant is 5.43 Angstrom. Figure 1. Diamond structure Sol) ρ = 8× 1 8 +6× 1 2 +4×1 # $ % & ' (×28.086 u atomic mass ! 5.43×10−10 ( ) 3 = 8×28.086×1.67×10−27 5.431×10−10 ( ) 3 = 2342.38kg m3 (1) Number density 8 atoms 5.431×10−8 ( ) 3 cm3 ~ 5×1022 atoms cm3 (2) Liedemann model (melting) Melting occurs when amplitude of vibrations exceeds critical fraction of interatomic spacing 1 2 gx0 2 = kB Tm (3) Tm: melting temperature. g as force constant. Let x0 as βa where a is the lattice constant. Typical number of β (critical fraction) in solid is 1/7. x0 = βa (4) Tm = 1 2 gx0 2 k B = 1 2 E ⋅ a⋅ x0 2 k B = 1 2 E ⋅ a3 ⋅β2 k B (5) where E is an elastic modulus of a solid.